Question 1

For the equation $\left( x ^ { 2 } - 16 \right) ^ { 3 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 0$ is

Accepted Answer

A)  an ordinary point 
B)  a regular singular point 
C)  an irregular singular point 
D)  a special point 
E)  none of the above 
A)  an ordinary point 
B)  a regular singular point 
C)  an irregular singular point 
D)  a special point 
E)  none of the above A

Question 2

For the differential equation $\left( x ^ { 2 } - 4 \right) ^ { 2 } y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 0$ is

Accepted Answer

A)  an ordinary point 
B)  a regular singular point 
C)  an irregular singular point 
D)  a special point 
E)  none of the above 
A)  an ordinary point 
B)  a regular singular point 
C)  an irregular singular point 
D)  a special point 
E)  none of the above A

Question 3

The recurrence relation for the differential equation $x y ^ { \prime \prime } + 2 y ^ { \prime } - x y = 0$ is

Accepted Answer

A)  $c _ { k } ( k + r ) ( k + r - 1 ) + c _ { k - 2 } = 0$ 
B)  $c _ { k } ( k + r ) ( k + r - 1 ) - c _ { k - 2 } = 0$ 
C)  $c _ { k } ( k + r + 1 ) ^ { 2 } - c _ { k - 2 } = 0$ 
D)  $c _ { k } ( k + r + 2 ) ( k + r + 1 ) + c _ { k - 2 } = 0$ 
E)  $c _ { k } ( k + r ) ( k + r + 1 ) - c _ { k - 2 } = 0$ 
A)  $c _ { k } ( k + r ) ( k + r - 1 ) + c _ { k - 2 } = 0$ 
B)  $c _ { k } ( k + r ) ( k + r - 1 ) - c _ { k - 2 } = 0$ 
C)  $c _ { k } ( k + r + 1 ) ^ { 2 } - c _ { k - 2 } = 0$ 
D)  $c _ { k } ( k + r + 2 ) ( k + r + 1 ) + c _ { k - 2 } = 0$ 
E)  $c _ { k } ( k + r ) ( k + r + 1 ) - c _ { k - 2 } = 0$ E

Question 4

The differential equation $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - 1 / 16 \right) y = 0$ is

Accepted Answer

A)  Bessel's equation of order $n$ 
B)  Bessel's equation of order 1/16 
C)  Bessel's equation of order 1/4 
D)  Legendre's equation of order 1/16 
E)  Legendre's equation of order 1/4 
A)  Bessel's equation of order $n$ 
B)  Bessel's equation of order 1/16 
C)  Bessel's equation of order 1/4 
D)  Legendre's equation of order 1/16 
E)  Legendre's equation of order 1/4

Question 5

The solution of the recurrence relation in the previous problem is

Accepted Answer

A)  $c _ { 2 k } = c _ { 0 } ( - 1 ) ^ { k } / ( 2 k ) , c _ { 2 k + 1 } = c _ { 1 } ( - 1 ) ^ { k } / ( 2 k + 1 )$ 
B)  $c _ { 2 k } = c _ { 0 } ( - 1 ) ^ { k } / ( 2 k ) ^ { 2 } , c _ { 2 k + 1 } = c _ { 1 } ( - 1 ) ^ { k } / ( 2 k + 1 ) ^ { 2 }$ 
C)  $c _ { 2 k } = c _ { 0 } ( - 1 ) ^ { k } / ( 2 k ) ! , c _ { 2 k + 1 } = c _ { 1 } ( - 1 ) ^ { k } / ( 2 k + 1 ) !$ 
D)  $c _ { 2 k } = c _ { 0 } ( - 1 ) ^ { k } / ( 2 k + 2 ) ! , c _ { 2 k + 1 } = c _ { 1 } ( - 1 ) ^ { k } / ( 2 k + 3 ) !$ 
E)  $c _ { 2 k } = c _ { 0 } ( - 1 ) ^ { k } / ( 2 k - 1 ) ! , c _ { 2 k + 1 } = c _ { 1 } ( - 1 ) ^ { k } / ( 2 k ) !$ 
A)  $c _ { 2 k } = c _ { 0 } ( - 1 ) ^ { k } / ( 2 k ) , c _ { 2 k + 1 } = c _ { 1 } ( - 1 ) ^ { k } / ( 2 k + 1 )$ 
B)  $c _ { 2 k } = c _ { 0 } ( - 1 ) ^ { k } / ( 2 k ) ^ { 2 } , c _ { 2 k + 1 } = c _ { 1 } ( - 1 ) ^ { k } / ( 2 k + 1 ) ^ { 2 }$ 
C)  $c _ { 2 k } = c _ { 0 } ( - 1 ) ^ { k } / ( 2 k ) ! , c _ { 2 k + 1 } = c _ { 1 } ( - 1 ) ^ { k } / ( 2 k + 1 ) !$ 
D)  $c _ { 2 k } = c _ { 0 } ( - 1 ) ^ { k } / ( 2 k + 2 ) ! , c _ { 2 k + 1 } = c _ { 1 } ( - 1 ) ^ { k } / ( 2 k + 3 ) !$ 
E)  $c _ { 2 k } = c _ { 0 } ( - 1 ) ^ { k } / ( 2 k - 1 ) ! , c _ { 2 k + 1 } = c _ { 1 } ( - 1 ) ^ { k } / ( 2 k ) !$

Question 6

Consider the differential equation $x y ^ { \prime \prime } - x y ^ { \prime } + y = 0$ . The indicial equation is $r ( r - 1 ) = 0$ . The recurrence relation is $c _ { k + 1 } ( k + r + 1 ) + ( k + r ) - c _ { k } ( k + r - 1 ) = 0$ . A series solution corresponding to the indicial root $r = 0$ is

Accepted Answer

A)  $y _ { 1 } = x$ 
B)  $y _ { 1 } = x ^ { 2 }$ 
C)  $y _ { 1 } = \sum _ { k = 0 } ^ { \infty } ( - 2 x ) ^ { k } / [ k ! ( - 1 ) \cdot 1 \cdot 3 \cdots ( 2 k - 1 ) ]$ 
D)  $y _ { 1 } = \sum _ { k = 0 } ^ { \infty } ( - 2 x ) ^ { k } / [ k ! ( 2 k - 3 ) ! ]$ 
E)  $y _ { 1 } = \sum _ { k = 0 } ^ { \infty } ( - 2 x ) ^ { k } / [ k ! 1 \cdot 3 \cdots ( 2 k - 3 ) ]$ 
A)  $y _ { 1 } = x$ 
B)  $y _ { 1 } = x ^ { 2 }$ 
C)  $y _ { 1 } = \sum _ { k = 0 } ^ { \infty } ( - 2 x ) ^ { k } / [ k ! ( - 1 ) \cdot 1 \cdot 3 \cdots ( 2 k - 1 ) ]$ 
D)  $y _ { 1 } = \sum _ { k = 0 } ^ { \infty } ( - 2 x ) ^ { k } / [ k ! ( 2 k - 3 ) ! ]$ 
E)  $y _ { 1 } = \sum _ { k = 0 } ^ { \infty } ( - 2 x ) ^ { k } / [ k ! 1 \cdot 3 \cdots ( 2 k - 3 ) ]$

Question 7

The solution of the previous problem is

Accepted Answer

A)  $y = c _ { 1 } P _ { 1 / 4 } ( x ) + c _ { 2 } P _ { - 1 / 4 } ( x )$ 
B)  $y = c _ { 1 } P _ { 4 } ( x ) + c _ { 2 } P _ { 4 } ( x )$ 
C)  $y = c _ { 1 } J _ { 4 } ( x ) + c _ { 2 } Y _ { 4 } ( x )$ 
D)  $y = c _ { 1 } J _ { 1 / 4 } ( x ) + c _ { 2 } J _ { - 1 / 4 } ( x )$ 
E)  $y = c _ { 1 } J _ { 1 / 16 } ( x ) + c _ { 2 } J _ { - 1 / 16 } ( x )$ 
A)  $y = c _ { 1 } P _ { 1 / 4 } ( x ) + c _ { 2 } P _ { - 1 / 4 } ( x )$ 
B)  $y = c _ { 1 } P _ { 4 } ( x ) + c _ { 2 } P _ { 4 } ( x )$ 
C)  $y = c _ { 1 } J _ { 4 } ( x ) + c _ { 2 } Y _ { 4 } ( x )$ 
D)  $y = c _ { 1 } J _ { 1 / 4 } ( x ) + c _ { 2 } J _ { - 1 / 4 } ( x )$ 
E)  $y = c _ { 1 } J _ { 1 / 16 } ( x ) + c _ { 2 } J _ { - 1 / 16 } ( x )$

Question 8

The differential equation is $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + 20 y = 0$ is

Accepted Answer

A)  Bessel's equation of order 20 
B)  Bessel's equation of order 4 
C)  Legendre's equation of order n 
D)  Legendre's equation of order 20 
E)  Legendre's equation of order 4 
A)  Bessel's equation of order 20 
B)  Bessel's equation of order 4 
C)  Legendre's equation of order n 
D)  Legendre's equation of order 20 
E)  Legendre's equation of order 4

Question 9

The solution of the previous problem is

Accepted Answer

A)  $y = c _ { 1 } P _ { 20 } ( x ) + c _ { 2 } P _ { - 20 } ( x )$ 
B)  $y = c _ { 1 } P _ { 4 } ( x ) + c _ { 2 } Q _ { 4 } ( x )$ , where $Q _ { 4 } ( x )$ is given by an infinite series 
C)  $y = c _ { 1 } J _ { 4 } ( x ) + c _ { 2 } Y _ { 4 } ( x )$ 
D)  $y = c _ { 1 } J _ { 1 / 4 } ( x ) + c _ { 2 } J _ { - 1 / 4 } ( x )$ 
E)  $y = c _ { 1 } J _ { 20 } ( x ) + c _ { 2 } Y _ { 20 } ( x )$ 
A)  $y = c _ { 1 } P _ { 20 } ( x ) + c _ { 2 } P _ { - 20 } ( x )$ 
B)  $y = c _ { 1 } P _ { 4 } ( x ) + c _ { 2 } Q _ { 4 } ( x )$ , where $Q _ { 4 } ( x )$ is given by an infinite series 
C)  $y = c _ { 1 } J _ { 4 } ( x ) + c _ { 2 } Y _ { 4 } ( x )$ 
D)  $y = c _ { 1 } J _ { 1 / 4 } ( x ) + c _ { 2 } J _ { - 1 / 4 } ( x )$ 
E)  $y = c _ { 1 } J _ { 20 } ( x ) + c _ { 2 } Y _ { 20 } ( x )$

Question 10

The singular points of the differential equation $x y ^ { \prime \prime } + y ^ { \prime } + y ( x + 2 ) / ( x - 4 ) = 0$ are

Accepted Answer

A)  none 
B)  0 
C)  0, $- 2$ 
D)  0, 4 
E)  0, $- 2$ , 4 
A)  none 
B)  0 
C)  0, $- 2$ 
D)  0, 4 
E)  0, $- 2$ , 4

Question 11

The recurrence relation for the power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } + y = 0$ is

Accepted Answer

A)  $( k + 2 ) ( k + 1 ) c _ { k + 2 } + c _ { k } = 0$ 
B)  $( k + 2 ) ( k + 1 ) c _ { k } + c _ { k - 2 } = 0$ 
C)  $( k + 1 ) k c _ { k + 2 } + c _ { k } = 0$ 
D)  $( k + 1 ) k c _ { k } + c _ { k - 2 } = 0$ 
E)  $( k - 2 ) ( k - 1 ) c _ { k - 2 } + c _ { k } = 0$ 
A)  $( k + 2 ) ( k + 1 ) c _ { k + 2 } + c _ { k } = 0$ 
B)  $( k + 2 ) ( k + 1 ) c _ { k } + c _ { k - 2 } = 0$ 
C)  $( k + 1 ) k c _ { k + 2 } + c _ { k } = 0$ 
D)  $( k + 1 ) k c _ { k } + c _ { k - 2 } = 0$ 
E)  $( k - 2 ) ( k - 1 ) c _ { k - 2 } + c _ { k } = 0$

Question 12

The recurrence relation for the power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } - y = 0$ is (for $k = 0,1,2 , \ldots$ )

Accepted Answer

A)  $( k + 2 ) ( k + 1 ) c _ { k + 2 } = c _ { k }$ 
B)  $( k + 2 ) ( k + 1 ) c _ { k } = c _ { k - 2 }$ 
C)  $( k + 1 ) k c _ { k + 2 } = c _ { k }$ 
D)  $( k + 1 ) k c _ { k } = c _ { k - 2 }$ 
E)  $( k - 2 ) ( k - 1 ) c _ { k - 2 } = c _ { k }$ 
A)  $( k + 2 ) ( k + 1 ) c _ { k + 2 } = c _ { k }$ 
B)  $( k + 2 ) ( k + 1 ) c _ { k } = c _ { k - 2 }$ 
C)  $( k + 1 ) k c _ { k + 2 } = c _ { k }$ 
D)  $( k + 1 ) k c _ { k } = c _ { k - 2 }$ 
E)  $( k - 2 ) ( k - 1 ) c _ { k - 2 } = c _ { k }$

Question 13

For the equation $\left( x ^ { 2 } - 16 \right) ^ { 3 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 1$ is

Accepted Answer

A)  an ordinary point 
B)  a regular singular point 
C)  an irregular singular point 
D)  a special point 
E)  none of the above 
A)  an ordinary point 
B)  a regular singular point 
C)  an irregular singular point 
D)  a special point 
E)  none of the above

Question 14

The differential equation is $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + 12 y = 0$ is

Accepted Answer

A)  Bessel's equation of order 12 
B)  Bessel's equation of order 3 
C)  Legendre's equation of order 12 
D)  Legendre's equation of order 3 
E)  Legendre's equation of order 4 
A)  Bessel's equation of order 12 
B)  Bessel's equation of order 3 
C)  Legendre's equation of order 12 
D)  Legendre's equation of order 3 
E)  Legendre's equation of order 4

Question 15

The solution of the recurrence relation in the previous problem is

Accepted Answer

A)  $c _ { 2 k } = c _ { 0 } / ( 2 k ) , c _ { 2 k + 1 } = c _ { 1 } / ( 2 k + 1 )$ 
B)  $c _ { 2 k } = c _ { 0 } / ( 2 k ) ^ { 2 } , c _ { 2 k + 1 } = c _ { 1 } / ( 2 k + 1 ) ^ { 2 }$ 
C)  $c _ { 2 k } = c _ { 0 } / ( 2 k ) ! , c _ { 2 k + 1 } = c _ { 1 } / ( 2 k + 1 ) !$ 
D)  $c _ { 2 k } = c _ { 0 } / ( 2 k + 2 ) ! , c _ { 2 k + 1 } = c _ { 1 } / ( 2 k + 3 ) !$ 
E)  $c _ { 2 k } = c _ { 0 } / ( 2 k - 1 ) ! , c _ { 2 k + 1 } = c _ { 1 } / ( 2 k ) !$ 
A)  $c _ { 2 k } = c _ { 0 } / ( 2 k ) , c _ { 2 k + 1 } = c _ { 1 } / ( 2 k + 1 )$ 
B)  $c _ { 2 k } = c _ { 0 } / ( 2 k ) ^ { 2 } , c _ { 2 k + 1 } = c _ { 1 } / ( 2 k + 1 ) ^ { 2 }$ 
C)  $c _ { 2 k } = c _ { 0 } / ( 2 k ) ! , c _ { 2 k + 1 } = c _ { 1 } / ( 2 k + 1 ) !$ 
D)  $c _ { 2 k } = c _ { 0 } / ( 2 k + 2 ) ! , c _ { 2 k + 1 } = c _ { 1 } / ( 2 k + 3 ) !$ 
E)  $c _ { 2 k } = c _ { 0 } / ( 2 k - 1 ) ! , c _ { 2 k + 1 } = c _ { 1 } / ( 2 k ) !$

Question 16

A power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } - y = 0$ is

Accepted Answer

A)  $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) ! + c _ { 1 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k + 1 } / ( 2 k + 1 ) !$ 
B)  $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) + c _ { 1 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k + 1 } / ( 2 k + 1 )$ 
C)  $y = c _ { 0 } \sum _ { k - 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) ^ { 2 } + c _ { 1 } \sum _ { k - 0 } ^ { \infty } x ^ { 2 k + 1 } / ( 2 k + 1 ) ^ { 2 }$ 
D)  $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) ! + c _ { 1 } \sum _ { k - 0 } ^ { \infty } x ^ { 2 k - 1 } / ( 2 k - 1 ) !$ 
E)  $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) + c _ { 1 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k - 1 } / ( 2 k - 1 )$ 
A)  $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) ! + c _ { 1 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k + 1 } / ( 2 k + 1 ) !$ 
B)  $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) + c _ { 1 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k + 1 } / ( 2 k + 1 )$ 
C)  $y = c _ { 0 } \sum _ { k - 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) ^ { 2 } + c _ { 1 } \sum _ { k - 0 } ^ { \infty } x ^ { 2 k + 1 } / ( 2 k + 1 ) ^ { 2 }$ 
D)  $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) ! + c _ { 1 } \sum _ { k - 0 } ^ { \infty } x ^ { 2 k - 1 } / ( 2 k - 1 ) !$ 
E)  $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) + c _ { 1 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k - 1 } / ( 2 k - 1 )$

Question 17

The solution of the previous problem is

Accepted Answer

A)  $y = c _ { 1 } P _ { 3 } ( x ) + c _ { 2 } P _ { - 3 } ( x )$ 
B)  $y = c _ { 1 } P _ { 3 } ( x ) + c _ { 2 } Q _ { 3 } ( x )$ , where $Q _ { 3 } ( x )$ is given by an infinite series 
C)  $y = c _ { 1 } J _ { 4 } ( x ) + c _ { 2 } Y _ { 4 } ( x )$ 
D)  $y = c _ { 1 } J _ { 3 } ( x ) + c _ { 2 } Y _ { 3 } ( x )$ 
E)  $y = c _ { 1 } J _ { 12 } ( x ) + c _ { 2 } Y _ { 12 } ( x )$ 
A)  $y = c _ { 1 } P _ { 3 } ( x ) + c _ { 2 } P _ { - 3 } ( x )$ 
B)  $y = c _ { 1 } P _ { 3 } ( x ) + c _ { 2 } Q _ { 3 } ( x )$ , where $Q _ { 3 } ( x )$ is given by an infinite series 
C)  $y = c _ { 1 } J _ { 4 } ( x ) + c _ { 2 } Y _ { 4 } ( x )$ 
D)  $y = c _ { 1 } J _ { 3 } ( x ) + c _ { 2 } Y _ { 3 } ( x )$ 
E)  $y = c _ { 1 } J _ { 12 } ( x ) + c _ { 2 } Y _ { 12 } ( x )$

Question 18

The singular points of $x ^ { 2 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ are $x =$ Select all that apply.

Accepted Answer

A)  2 
B)  $- 1$ 
C)  0 
D)  1 
E)  none of the above 
A)  2 
B)  $- 1$ 
C)  0 
D)  1 
E)  none of the above

Question 19

In the previous problem, a second solution is

Accepted Answer

A)  $y _ { 2 } = e ^ { x }$ 
B)  $y _ { 2 } = x \int e ^ { x } / x ^ { 2 } d x$ 
C)  $y = 1 + \sum _ { k - 1 } ^ { \infty } c _ { k } x ^ { k } , \text { where } c _ { k } = ( k - 1 ) / ( k ( k + 1 ) )$ 
D)  $y = 1 + \sum _ { k = 1 } ^ { \infty } c _ { k } x ^ { k } , \text { where } c _ { k } = 1 / k ^ { 2 }$ 
E)  none of the above 
A)  $y _ { 2 } = e ^ { x }$ 
B)  $y _ { 2 } = x \int e ^ { x } / x ^ { 2 } d x$ 
C)  $y = 1 + \sum _ { k - 1 } ^ { \infty } c _ { k } x ^ { k } , \text { where } c _ { k } = ( k - 1 ) / ( k ( k + 1 ) )$ 
D)  $y = 1 + \sum _ { k = 1 } ^ { \infty } c _ { k } x ^ { k } , \text { where } c _ { k } = 1 / k ^ { 2 }$ 
E)  none of the above

Question 20

For the equation $\left( x ^ { 2 } - 16 \right) ^ { 3 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 4$ is

Accepted Answer

A)  an ordinary point 
B)  a regular singular point 
C)  an irregular singular point 
D)  a special point 
E)  none of the above 
A)  an ordinary point 
B)  a regular singular point 
C)  an irregular singular point 
D)  a special point 
E)  none of the above

For the equation $\left( x ^ { 2 } - 16 \right) ^ { 3 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 0$ is

For the differential equation $\left( x ^ { 2 } - 4 \right) ^ { 2 } y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 0$ is

The recurrence relation for the differential equation $x y ^ { \prime \prime } + 2 y ^ { \prime } - x y = 0$ is

The differential equation $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - 1 / 16 \right) y = 0$ is

The solution of the recurrence relation in the previous problem is

Consider the differential equation $x y ^ { \prime \prime } - x y ^ { \prime } + y = 0$ . The indicial equation is $r ( r - 1 ) = 0$ . The recurrence relation is $c _ { k + 1 } ( k + r + 1 ) + ( k + r ) - c _ { k } ( k + r - 1 ) = 0$ . A series solution corresponding to the indicial root $r = 0$ is

The solution of the previous problem is

The differential equation is $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + 20 y = 0$ is

The solution of the previous problem is

The singular points of the differential equation $x y ^ { \prime \prime } + y ^ { \prime } + y ( x + 2 ) / ( x - 4 ) = 0$ are

The recurrence relation for the power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } + y = 0$ is

The recurrence relation for the power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } - y = 0$ is (for $k = 0,1,2 , \ldots$ )

For the equation $\left( x ^ { 2 } - 16 \right) ^ { 3 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 1$ is

The differential equation is $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + 12 y = 0$ is

The solution of the recurrence relation in the previous problem is

A power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } - y = 0$ is

The solution of the previous problem is

The singular points of $x ^ { 2 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ are $x =$ Select all that apply.

In the previous problem, a second solution is

For the equation $\left( x ^ { 2 } - 16 \right) ^ { 3 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 4$ is

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

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Exam 6: Series Solutions of Linear Equations

For the equation (x2−16)3(x−1)y′′−2xy′+y=0\left( x ^ { 2 } - 16 \right) ^ { 3 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0(x2−16)3(x−1)y′′−2xy′+y=0 , the point x=0x = 0x=0 is

For the differential equation (x2−4)2y′′−2xy′+y=0\left( x ^ { 2 } - 4 \right) ^ { 2 } y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0(x2−4)2y′′−2xy′+y=0 , the point x=0x = 0x=0 is

The recurrence relation for the differential equation xy′′+2y′−xy=0x y ^ { \prime \prime } + 2 y ^ { \prime } - x y = 0xy′′+2y′−xy=0 is

The differential equation x2y′′+xy′+(x2−1/16)y=0x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - 1 / 16 \right) y = 0x2y′′+xy′+(x2−1/16)y=0 is

The solution of the recurrence relation in the previous problem is

The solution of the previous problem is

The differential equation is (1−x2)y′′−2xy′+20y=0\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + 20 y = 0(1−x2)y′′−2xy′+20y=0 is

The solution of the previous problem is

The singular points of the differential equation xy′′+y′+y(x+2)/(x−4)=0x y ^ { \prime \prime } + y ^ { \prime } + y ( x + 2 ) / ( x - 4 ) = 0xy′′+y′+y(x+2)/(x−4)=0 are

The recurrence relation for the power series solution about x=0x = 0x=0 of the differential equation y′′+y=0y ^ { \prime \prime } + y = 0y′′+y=0 is

The recurrence relation for the power series solution about x=0x = 0x=0 of the differential equation y′′−y=0y ^ { \prime \prime } - y = 0y′′−y=0 is (for k=0,1,2,…k = 0,1,2 , \ldotsk=0,1,2,… )

For the equation (x2−16)3(x−1)y′′−2xy′+y=0\left( x ^ { 2 } - 16 \right) ^ { 3 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0(x2−16)3(x−1)y′′−2xy′+y=0 , the point x=1x = 1x=1 is

The differential equation is (1−x2)y′′−2xy′+12y=0\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + 12 y = 0(1−x2)y′′−2xy′+12y=0 is

The solution of the recurrence relation in the previous problem is

A power series solution about x=0x = 0x=0 of the differential equation y′′−y=0y ^ { \prime \prime } - y = 0y′′−y=0 is

The solution of the previous problem is

The singular points of x2(x−1)y′′−2xy′+y=0x ^ { 2 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0x2(x−1)y′′−2xy′+y=0 are x=x =x= Select all that apply.

In the previous problem, a second solution is

For the equation (x2−16)3(x−1)y′′−2xy′+y=0\left( x ^ { 2 } - 16 \right) ^ { 3 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0(x2−16)3(x−1)y′′−2xy′+y=0 , the point x=4x = 4x=4 is

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

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For the equation $\left( x ^ { 2 } - 16 \right) ^ { 3 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 0$ is

For the differential equation $\left( x ^ { 2 } - 4 \right) ^ { 2 } y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 0$ is

The recurrence relation for the differential equation $x y ^ { \prime \prime } + 2 y ^ { \prime } - x y = 0$ is

The differential equation $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - 1 / 16 \right) y = 0$ is

The differential equation is $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + 20 y = 0$ is

The singular points of the differential equation $x y ^ { \prime \prime } + y ^ { \prime } + y ( x + 2 ) / ( x - 4 ) = 0$ are

The recurrence relation for the power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } + y = 0$ is

The recurrence relation for the power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } - y = 0$ is (for $k = 0,1,2 , \ldots$ )

For the equation $\left( x ^ { 2 } - 16 \right) ^ { 3 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 1$ is

The differential equation is $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + 12 y = 0$ is

A power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } - y = 0$ is

The singular points of $x ^ { 2 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ are $x =$ Select all that apply.

For the equation $\left( x ^ { 2 } - 16 \right) ^ { 3 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 4$ is